1. Nozzle Theory¶

This section derives the compressible flow relations used throughout machwave for nozzle flow modelling. Every result is traced back to two fundamental laws: the First Law of Thermodynamics and the ideal gas law.

Assumptions

Quasi-one-dimensional flow (flow properties uniform across every cross-section).
Adiabatic, inviscid — no heat exchange or friction inside the nozzle.
Calorically perfect gas: specific heats \(c_p\), \(c_v\) and their ratio \(k = c_p/c_v\) are constant.
No shaft work.

1.1 Steady-Flow Energy Equation¶

For a steady, adiabatic, work-free flow the First Law requires the specific stagnation enthalpy to be conserved along every streamline:

\[ \boxed{h + \frac{v^2}{2} = \text{constant}} \]

Applying this between any two stations \(x\) and \(y\):

\[ h_x + \frac{v_x^2}{2} = h_y + \frac{v_y^2}{2} \]

which rearranges to

\[ h_x - h_y = \frac{1}{2}\left(v_y^2 - v_x^2\right). \]

For a calorically perfect gas \(h = c_p T\), so this becomes

\[ c_p\left(T_x - T_y\right) = \frac{1}{2}\left(v_y^2 - v_x^2\right). \]

Choosing station \(x\) as the stagnation state (\(v = 0\), subscript \(0\)) and \(y\) as any point in the flow:

\[ \boxed{c_p T_0 = c_p T + \frac{v^2}{2}} \]

\(T_0\) is the stagnation temperature, a conserved quantity along the flow.

The ideal gas relation \(c_p - c_v = R\) and \(k = c_p/c_v\) give

\[ c_p = \frac{kR}{k-1}, \qquad R = \frac{\bar{R}}{M_w} \]

where \(\bar{R} = 8.314\ \text{J mol}^{-1}\text{K}^{-1}\) and \(M_w\) is the molar mass of the combustion products.

1.2 Mach Number and Stagnation Temperature Ratio¶

The local speed of sound in a perfect gas is \(a = \sqrt{kRT}\). Defining the Mach number \(M = v/a\) and substituting \(v = Ma\) into the energy equation:

\[ c_p T_0 = c_p T + \frac{(Ma)^2}{2} = c_p T + \frac{kRM^2T}{2} = c_p T\left(1 + \frac{k-1}{2}M^2\right) \]

since \(\frac{kR}{2c_p} = \frac{k-1}{2}\). Therefore

\[ \boxed{\frac{T_0}{T} = 1 + \frac{k-1}{2}M^2} \]

1.3 Isentropic Stagnation Relations¶

For an isentropic process the entropy is constant, which for a perfect gas means

\[ P\rho^{-k} = \text{const}, \qquad \frac{P}{P_0} = \left(\frac{\rho}{\rho_0}\right)^k. \]

Combined with the ideal gas law \(P = \rho R T\):

\[ \frac{P}{P_0} = \left(\frac{T}{T_0}\right)^{k/(k-1)}, \qquad \frac{\rho}{\rho_0} = \left(\frac{T}{T_0}\right)^{1/(k-1)}. \]

Substituting the temperature ratio from §1.2:

\[ \boxed{\frac{P_0}{P} = \left(1 + \frac{k-1}{2}M^2\right)^{k/(k-1)}} \qquad \boxed{\frac{\rho_0}{\rho} = \left(1 + \frac{k-1}{2}M^2\right)^{1/(k-1)}} \]

These are the isentropic stagnation relations, implemented in get_exit_pressure.

1.4 Critical (Sonic) Conditions and Choked Flow¶

At the throat \(M = 1\). Evaluating the stagnation relations at this condition (denoting the sonic state with \(^*\)):

\[ \frac{T^*}{T_0} = \frac{2}{k+1} \]

\[ \boxed{\frac{P^*}{P_0} = \left(\frac{2}{k+1}\right)^{k/(k-1)}} \]

\[ \frac{\rho^*}{\rho_0} = \left(\frac{2}{k+1}\right)^{1/(k-1)} \]

The nozzle is choked — the throat is sonic and the mass flow rate has reached its maximum — whenever the back pressure satisfies

\[ \frac{P_\text{ext}}{P_0} \leq \left(\frac{2}{k+1}\right)^{k/(k-1)}. \]

Implemented in get_critical_pressure_ratio and is_flow_choked.

1.5 Area–Mach Relation¶

Applying continuity between the throat and an arbitrary cross-section of area \(A\):

\[ \rho v A = \rho^* a^* A_t \implies \frac{A}{A_t} = \frac{\rho^* a^*}{\rho v}. \]

Expressing each factor using §1.2 and §1.3:

\[ \frac{a^*}{v} = \frac{a^*}{Ma} = \frac{1}{M}\sqrt{\frac{T^*}{T}} = \frac{1}{M}\sqrt{\frac{2}{k+1}\cdot\frac{1}{1+\frac{k-1}{2}M^2}} \]

\[ \frac{\rho^*}{\rho} = \left(\frac{T^*}{T_0}\cdot\frac{T_0}{T}\right)^{1/(k-1)} = \left(\frac{2}{(k+1)}\left(1+\frac{k-1}{2}M^2\right)\right)^{1/(k-1)} \]

Combining:

\[ \boxed{\frac{A}{A_t} = \frac{1}{M}\left[\frac{2}{k+1}\left(1+\frac{k-1}{2}M^2\right)\right]^{(k+1)/[2(k-1)]}} \]

For a given expansion ratio \(\varepsilon = A_e/A_t\) there are two solutions: \(M < 1\) (subsonic) and \(M > 1\) (supersonic). machwave always selects the supersonic root using Brent's method; see get_exit_mach_from_expansion_ratio.

1.6 Exit Velocity¶

Applying the energy equation (§1.1) between the chamber (\(v \approx 0\)) and the nozzle exit:

\[ v_e = \sqrt{2c_p T_0\left[1 - \frac{T_e}{T_0}\right]} = \sqrt{\frac{2kR}{k-1}T_0\left[1 - \left(\frac{P_e}{P_0}\right)^{(k-1)/k}\right]} \]

where the last step uses the isentropic temperature ratio from §1.3 with \(T_e/T_0 = (P_e/P_0)^{(k-1)/k}\).

1.7 Choked Mass Flow Rate¶

At the throat \(M = 1\), so \(v^* = a^* = \sqrt{kRT^*}\). Using the continuity equation and the critical relations from §1.4:

\[ \dot{m} = \rho^* a^* A_t = \frac{\rho_0}{\left(\frac{\rho_0}{\rho^*}\right)}\sqrt{kRT^*}\, A_t \]

Substituting \(\rho_0 = P_0/(RT_0)\), \(T^* = 2T_0/(k+1)\):

\[ \boxed{\dot{m} = \frac{P_0\, A_t}{\sqrt{T_0}}\sqrt{\frac{k}{R}}\left(\frac{2}{k+1}\right)^{(k+1)/[2(k-1)]}} \]

This is the key expression linking chamber pressure to propellant mass flow — used in all mass-balance ODEs throughout machwave.

1.8 Thrust Coefficient¶

The thrust on a rocket nozzle equals the momentum flux leaving the exit plus the pressure difference acting on the exit area:

\[ F = \dot{m}\,v_e + (P_e - P_\text{ext})\,A_e. \]

Dividing by \(P_0 A_t\) defines the dimensionless thrust coefficient \(C_f = F / (P_0 A_t)\). Substituting \(\dot{m}\) from §1.7 and \(v_e\) from §1.6:

\[ \frac{\dot{m}\,v_e}{P_0 A_t} = \sqrt{\frac{k}{R}\left(\frac{2}{k+1}\right)^{(k+1)/(k-1)}}\cdot \sqrt{\frac{2kR}{k-1}\left[1-\left(\frac{P_e}{P_0}\right)^{(k-1)/k}\right]} = \sqrt{\frac{2k^2}{k-1}\left(\frac{2}{k+1}\right)^{(k+1)/(k-1)}\left[1-\left(\frac{P_e}{P_0}\right)^{(k-1)/k}\right]} \]

Therefore:

\[ \boxed{C_f = \sqrt{\frac{2k^2}{k-1}\left(\frac{2}{k+1}\right)^{(k+1)/(k-1)}\!\left[1-\left(\frac{P_e}{P_0}\right)^{(k-1)/k}\right]} + \varepsilon\,\frac{P_e - P_\text{ext}}{P_0}} \]

with \(\varepsilon = A_e/A_t\). Thrust follows immediately from the definition:

\[ F = C_f\, P_0\, A_t. \]

Implemented in get_ideal_thrust_coefficient and get_thrust_from_thrust_coefficient.

1.9 Optimal Expansion Ratio¶

The pressure-thrust term \(\varepsilon(P_e - P_\text{ext})/P_0\) vanishes when \(P_e = P_\text{ext}\) (perfectly expanded nozzle). Inserting \(P_e = P_\text{ext}\) into the area–Mach relation (§1.5), with the exit Mach determined from the isentropic pressure ratio (§1.3):

\[ \varepsilon_\text{opt} = \frac{1}{M_e} \left[\frac{2}{k+1}\left(1+\frac{k-1}{2}M_e^2\right)\right]^{(k+1)/[2(k-1)]} \]

where \(M_e\) satisfies \(P_0/P_\text{ext} = (1 + (k-1)M_e^2/2)^{k/(k-1)}\), or equivalently in closed form:

\[ \boxed{\varepsilon_\text{opt} = \left[ \left(\frac{k+1}{2}\right)^{1/(k-1)} \left(\frac{P_\text{ext}}{P_0}\right)^{1/k} \sqrt{\frac{k+1}{k-1}\left(1 - \left(\frac{P_\text{ext}}{P_0}\right)^{(k-1)/k}\right)} \right]^{-1}} \]

Implemented in get_optimal_expansion_ratio.

References¶

Anderson, J. D. (2003). Modern Compressible Flow: With Historical Perspective (3rd ed.). McGraw-Hill.
Sutton, G. P., & Biblarz, O. (2017). Rocket Propulsion Elements (9th ed.). Wiley. Ch. 3.